I have a chord progression that goes:
A Bm D A G
Other areas of the song also have an A, G, D, A progression.
Would this song considered to be in A mixolydian vs A major or D major since the root seems to A, and the 7th is a flat 7th?
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The song definitely has the more an A Mixolydian feel then a D major feel. Both contain similar chord progressions, but there are a few signs showing A Mixolydian is the better way to look at it.
First of all, the progression itself centers around A as the tonic. Also note that the dominant chord (E major) is not present in this progression which would very strongly suggest an A major tonally. The VII chord (G major) is very present and used in a very Mixolydian way.
Let's examine the Roman Numeral Analysis that would be associated with each mode choice.
In D major the results would be
V vi I V IV
V IV I V
Not unheard progressions, but if we instead analyze it in A Mixolydian we get:
I ii IV I VII
I VII IV I
This analysis shows how the progression centers more around A then D thus makes more sense to call it A Mixolydian.
All the notes involved are present in A mixolydian, which is the 5th mode of D major = D ionian. The two scales are relative to each other: they share the same notes. If it was a modal progression, staying in A7 for most of the time, you could say it was in A mixolydian. But that's not the case.
There is a G# in A major, not a G, so that's wrong too.
It would be more appropriate to say this chord progression is in D major. It is not that common to consider a mode as the key to a piece, unless it clearly has a modal flavor. We use the parent major scale for that.
Let's take a closer look:
As you can see, there is little difference between the two chords. In fact, Bm can be a voicing for a D6 chord! The vi chord can be a substitute for the I chord.
This also happens with the G:
The happens here. The IV chord can be a substitute for the ii chord.
Because of the notes they share, the chords can be used to reharmonize progressions. Take a look at this Berklee course.
If you consider reharmonization, your original progression could be:
V I V ii
This has a clear D major feel, and it can be what your musical ear is actually perceiving.
It's very common to use chords 'borrowed' from the parallel. As in A maj - with A, Bm, C#m, D, E, F#m and ...Am, C, Dm, E (Em), F, G. Yes I left out the 'dim'.
Doing this in a song doesn't put it into a mode, but it still retains its original letter name as its key.
There is no mention about a G# NOTE in the song, at a different place. If that exists, then it can't be A Mixolydian, which is exactly the same note make up as D major.If every G is G natural, and the song gravitates to A, it could well be construed as A Mix.